On-line estimation method of battery state of health in wide temperature range based on “standardized temperature”

ABSTRACT

An on-line State of Health estimation method of a battery in a wide temperature range based on “standardized temperature” includes: calculating battery Incremental Capacity curve of a battery, establishing a quantitative relationship between the voltage shift of the temperature-sensitive feature point and the temperature of a standard battery, standardized transformation of Incremental Capacity curves at different temperatures, establishing a quantitative relationship between the transformed height of the capacity-sensitive feature point and the State of Health based on a BOX-COX transformation. The BOX-COX transformation is expressed as 
               y   k     (   λ   )       =     {                 y   k   λ     -   1     λ           λ   ≠   0               ln   ⁢           ⁢     y   k             λ   =   0           .             
An maximum likelihood function is used to calculate the optimal λ, and the transformed height of the capacity-sensitive feature point y can be acquired. The quantitative relationship between transformed height of the capacity-sensitive feature point and the State of Health is established to obtain the State of Health.

CROSS REFERENCE TO THE RELATED APPLICATIONS

This application is the national phase entry of InternationalApplication No. PCT/CN2021/070228, filed on Jan. 5, 2021, which is basedupon and claims priority to Chinese Patent Application No.202010468280.4, filed on May 28, 2020, the entire contents of which areincorporated herein by reference.

TECHNICAL FIELD

The present invention belongs to the electric vehicle technology,specifically relates to the State of Health estimation of a battery.

BACKGROUND

Accurate State of Health estimation is of great significance forimproving battery safety and extending battery life. However, due to thecomplicated working conditions and various influencing factors in theactual conditions, it is difficult to estimate the State of Healthaccurately and effectively.

State of Health is generally estimated by a maximum available capacitymethod or an internal resistance method. At present, battery State ofHealth is mostly obtained by establishing models, such as anelectrochemical model, an empirical model and an equivalent circuitmodel. Though the calculation results of the electrochemical model areextremely accurate, the model parameters of the electrochemical modelare too numerous to obtain and the calculation process is verycomplicated. The empirical model requires a large amount of test dataand takes a long time, resulting in a poor versatility. The equivalentcircuit model equates a battery by some resistors and capacitors with aseries-parallel structure. Although the State of Health is monitored inreal time, the accuracy is not high, and multiple optimizationalgorithms are usually required.

An incremental capacity curve is obtained by deriving a charging curveof a battery. The incremental capacity curve directly reflects themulti-stage lithium ions intercalation process when the battery ischarged. It is a non-destructive method to detect the internal reactionof the battery. By studying the relationship between the inflectionpoints of the Incremental Capacity curve and the State of Health at astandard temperature, a State of Health estimation method by theIncremental Capacity curve is proposed. However, the internal chemicalreaction of a battery is directly affected by the temperature, so manybasic battery parameters, such as the capacity and internal resistance,will change with the temperature. Meanwhile, the Incremental Capacitycurve will also shift with the temperature. The accuracy of the State ofHealth estimation method by the Incremental Capacity curve variesgreatly at different temperatures. Therefore, a State of Healthestimation method in a wide temperature range needs to be explored.

SUMMARY

To solve the above problems, based on “standardized temperature”, anonline State of Health estimation method in a wide temperature range isproposed. It mainly includes calculating Incremental Capacity curve of abattery, establishing a quantitative relationship between the voltageshift of the temperature-sensitive feature point and the temperature ofa standard battery, standardized transformation of Incremental Capacitycurves at different temperatures and establishing a quantitativerelationship between the transformed height of the capacity-sensitivefeature point and the State of Health of the battery based on a BOX-COXtransformation.

The mentioned Incremental Capacity curve is often calculated by aconventional numerical differentiation method, and by a polynomialfitting and differentiation method. It can also be calculated byreferring to the invention patent CN 109632138 A. The present inventiononly provides examples, the specific calculated method is not limited.

The method of establishing the quantitative relationship between thevoltage shift of the temperature-sensitive feature point and thetemperature of a standard battery is as follows:

The internal resistance of a battery increases as the temperaturedecreases, resulting in the increase of kinetic loss. The increase ofinternal resistance is mainly manifested as the right shift of theIncremental Capacity curve and the voltage shift of inflection points inthe Incremental capacity curve, as shown in FIG. 1. Therefore, thesecond inflection point in the Incremental Capacity curve is selected asthe temperature-sensitive feature point to establish the quantitativerelationship between the voltage shift of the temperature-sensitivefeature point and the temperature. Firstly, two standard batteries, newbatteries, are selected to stand for 2 hours at each fixed ambienttemperature (−5° C., 0° C., 5° C., 10° C., 15° C., 20° C., 25° C., 30°C., 35° C., 40° C., 45° C., 50° C., 55° C.) to ensure the consistenttemperature inside and outside the standard batteries. Then, thestandard batteries are charged with 0.1 C, and the Incremental Capacitycurves at different temperatures are extracted from the charging curves.Subsequently, the voltages of the temperature-sensitive feature point ateach temperature are recorded. To describe the quantitative relationshipbetween voltage shift of the temperature-sensitive feature and thetemperature, taking the standard temperature (25° C.) as a reference,the quantitative relationship between the voltage shift of thetemperature-sensitive feature point and the temperature is obtained bysubtracting voltage of the temperature-sensitive feature point at thestandard temperature from the voltage of the temperature-sensitivefeature point at other temperatures. And an Arrhenius fitting functionis utilized to fit the quantitative relationship:

$\begin{matrix}{y = {{a\;{\exp\left( \frac{b}{T} \right)}} + c}} & (1)\end{matrix}$

Where a, b, c are the fitting parameters, T is the temperature, and yrepresents the voltage shift of temperature-sensitive feature point.

The above standardized transformation of Incremental Capacity curves atdifferent temperatures is as follows:

The Incremental Capacity curves at different temperatures arestandardized to ensure the accurate estimation of State of Health in awide range temperature. According to the quantitative relationshipbetween the voltage shift of the temperature-sensitive feature point andthe temperature by Arrhenius fitting function, a charging Q-V curve ateach high temperature (upper 30° C.) is shifted with the correspondingvoltage shift (bringing the temperature into the Arrhenius fittingequation) to get a normalized charging Q-V curve. Then, a normalizedIncremental Capacity curve by standardized temperature is obtainedthrough the conventional numerical differentiation method.

Based on a BOX-COX transformation, the quantitative relationship betweenthe transformed height of the capacity-sensitive feature point and theState of Health of the battery is established as follows:

As the battery ages, battery's cathode and anode active materials andrecyclable lithium ions are gradually lost. They are mainly reflected inthe height decrease of the Incremental Capacity curve, as shown inFIG. 1. Therefore, the height of the second inflection point is used asthe capacity-sensitive feature point.

In this method, the Box-Cox transformation is used to increase thelinearity between the height of the capacity-sensitive feature point andthe State of Health. A linear regression equation is expressed as:

$\begin{matrix}{Y = {{X\;\beta} + ɛ}} & (2)\end{matrix}$

Where, Y is the dependent variable, X is the independent variable, β isthe coefficient matrix, and ε represents the fitting error.

The Box-Cox transformation is expressed as:

$\begin{matrix}{y_{k}^{(\lambda)} = \left\{ \begin{matrix}\frac{y_{k}^{\lambda} - 1}{\lambda} & {\lambda \neq 0} \\{\ln\; y_{k}} & {\lambda = 0}\end{matrix} \right.} & (3)\end{matrix}$

Where, y in the right side of the equation is the original variable, andthe subscript k corresponding toy represents the k-th variable. λ is theconversion parameter. y_(k) ^((λ)) in the left side of the equation isthe k-th conversion variable.

According to the above formula, the inverse transformation of y is:

$\begin{matrix}{y_{k} = \left\{ \begin{matrix}\left( {1 - {\lambda\; y_{k}^{(\lambda)}}} \right)^{1/\lambda} & {\lambda \neq 0} \\{\exp\left( y_{k}^{(\lambda)} \right)} & {\lambda = 0}\end{matrix} \right.} & (4)\end{matrix}$

A maximum likelihood function is used to calculate the optimal λ.Assuming that ε is independent and obeys normal distribution, and yconforms to y˜(Xβ,σ²I), X is the independent variable matrix, β is thecoefficient matrix, σ² is the variance, I is the identity matrix, n isthe number of samples. The maximum likelihood function is expressed as:

$\begin{matrix}{{L\left( {\beta,{\sigma^{2}❘Y},X} \right)} = {{{- \frac{n}{2}}\ln\; 2\pi} - {\frac{n}{2}\ln\;\sigma^{2}} - {\frac{1}{2}{{\sigma^{2}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}} + {\left( {\lambda - 1} \right){\sum\limits_{k = 1}^{n}{\ln\; y_{k}}}}}} & (5)\end{matrix}$

In the above formula (5), Y^((λ)) is the dependent variable afterconversion. The estimated parameters β and σ² can be expressed as:

$\begin{matrix}\left\{ \begin{matrix}{{\beta(\lambda)} = {\left( {X^{T}X} \right)^{- 1}X^{T}Y^{\lambda}}} \\{{\sigma^{2}(\lambda)} = {{\frac{1}{n}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}}\end{matrix} \right. & (6)\end{matrix}$

Substituting formula (6) into (5) and taking logarithmic:

$\begin{matrix}{{L(\lambda)} = {{{- \frac{n}{2}}\left( {{\ln\; 2\pi} + {\ln\;\sigma^{2}} + 1} \right)} + {\left( {\lambda - 1} \right){\overset{n}{\sum\limits_{k = 1}}{\ln\; y_{k}}}}}} & (7)\end{matrix}$

The optimal λ is obtained by maximizing formula (7).

According to the linear fitting formula (2), the quantitativerelationship between transformed height of capacity-sensitive featurepoint and the State of Health is established.

The Benefits of the Invention:

1. According to standardized temperature, the temperature range of theState of Health estimation method by the Incremental Capacity curve iseffectively broadened. The accuracy of the State of Health estimation bythe Incremental Capacity curve in a wide temperature range is alsoimproved by “standardized temperature” transformation.

2. The Box-Cox transformation is introduced to reduce the influence ofrandom errors on the estimation result, increase the linearity betweenthe height of the capacity-sensitive feature point and the State ofHealth, and improves the stability of the State of Health estimation bythe Incremental Capacity curve.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the Incremental Capacity curves at different State ofHealth and temperature.

FIG. 2 gives the procedure of the proposed State of Health estimationmethod.

FIG. 3 shows the fitting curve of the voltage shifts with temperatures.

FIG. 4-FIG. 9 shows the normalized Incremental Capacity curve at 30° C.,35° C., 40° C., 45° C., 50° C. and 55° C. respectively.

FIG. 10 shows the fitting results of State of Health and the transformedheight of the capacity-sensitive feature point.

DETAILED DESCRIPTION OF THE EMBODIMENTS

It will be further explained below about the figures.

FIG. 1 gives the Incremental Capacity curves at different State ofHealth and temperature. When the temperature decreases, the IncrementalCapacity curve has a right shift, and the height of inflection pointdecreases as the battery ages. But the height of the inflection point isalso affected by the voltage, so the voltage shift caused by temperatureis excluded by temperature standardized transformation.

FIG. 2 gives the procedure of the proposed State of Health estimation inthe invention. It mainly includes two parts, an offline calibrationstage and an online estimation stage.

The State of Health of the selected five batteries are shown in Table.1.

TABLE 1 Battery No. 1 No. 2 No. 3 No. 4 No. 5 State of Health 0.98 0.880.82 0.70 0.56

In the offline calibration stage, the standard batteries, new batteries,are stand for 2 hours at each fixed ambient temperature (−5° C., 0° C.,5° C., 10° C., 15° C., 20° C., 25° C., 30° C., 35° C., 40° C., 45° C.,50° C., 55° C.) to ensure the consistent temperature inside and outsidethe standard batteries. Then, the standard batteries are charged with0.1 C, and the Incremental Capacity curves at different temperatures areextracted from the charging curves by numerical calculation. Thequantitative relationship between the voltage shift of thetemperature-sensitive feature point and the temperature is obtained bysubtracting the voltage of the temperature-sensitive feature point atthe standard temperature from the voltage of the temperature-sensitivefeature point at other temperatures. Then, the Arrhenius fittingfunction is utilized to fit the quantitative relationship. The fittingresults are shown in FIG. 3, and the parameters a=3.66328E−12,b=6300·19841, c=−0.00621.

According to the fitting results, the charging curve at each hightemperature (upper 30° C.) is shifted with the corresponding voltageshift to get the normalized charging Q-V curve. Then, the normalizedIncremental Capacity curve at standardized temperature is obtained bythe conventional numerical differentiation method, by the polynomialfitting and differentiation method or by referring to the inventionpatent CN 109632138 A. FIG. 4-FIG. 9 are shows the normalizedIncremental Capacity curves under 30° C., 35° C., 40° C., 45° C., 50°C., 55° C., respectively.

After that, the height of the capacity-sensitive feature point isextracted and transformed by the Box-Cox method.

The Box-Cox transformation is expressed as:

$\begin{matrix}{y_{k}^{(\lambda)} = \left\{ \begin{matrix}\frac{y_{k}^{\lambda} - 1}{\lambda} & {\lambda \neq 0} \\{\ln\; y_{k}} & {\lambda = 0}\end{matrix} \right.} & (8)\end{matrix}$

Where, y in the right side of the equation is the original variable, andthe subscript k corresponding to y represents the k-th variable. λ isthe calculated conversion parameter. y_(k) ^((λ)) in the left side ofthe equation is the k-th conversion variable.

According to the above formula, the inverse transformation of y is:

$\begin{matrix}{y_{k} = \left\{ \begin{matrix}\left( {1 - {\lambda\; y_{k}^{(\lambda)}}} \right)^{1/\lambda} & {\lambda \neq 0} \\{\exp\left( y_{k}^{(\lambda)} \right)} & {\lambda = 0}\end{matrix} \right.} & (9)\end{matrix}$

The maximum likelihood function is used to calculate the optimal λ.Assuming that ε is independent and obeys normal distribution, and yconforms to y˜(Xβ,σ²I) X is the independent variable matrix, β is thecoefficient matrix, σ² is the variance, I is the identity matrix, themaximum likelihood function is expressed as:

$\begin{matrix}{{L\left( {\beta,{\sigma^{2}❘Y},X} \right)} = {{{- \frac{n}{2}}\ln\; 2\pi} - {\frac{n}{2}\ln\;\sigma^{2}} - {\frac{1}{2}{{\sigma^{2}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}} + {\left( {\lambda - 1} \right){\sum\limits_{k = 1}^{n}{\ln\; y_{k}}}}}} & (10)\end{matrix}$

Where, n is the total number of samples. The estimated parameters β andσ² can be expressed as:

$\begin{matrix}\left\{ \begin{matrix}{{\beta(\lambda)} = {\left( {X^{T}X} \right)^{- 1}X^{T}Y^{\lambda}}} \\{{\sigma^{2}(\lambda)} = {\frac{1}{n}{{\sigma^{2}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}}}\end{matrix} \right. & (11)\end{matrix}$

Substituting formula (11) into (10) and taking logarithmic:

$\begin{matrix}{{L(\lambda)} = {{{- \frac{n}{2}}\left( {{\ln\; 2\pi} + {\ln\;\sigma^{2}} + 1} \right)} + {\left( {\lambda - 1} \right){\sum\limits_{k = 1}^{n}{\ln\; y_{k}}}}}} & (12)\end{matrix}$

Taking its maximum value to get λ=3.3, and bringing it into formula (8)to get the transformed height of the capacity-sensitive feature point.

The relationship between the transformed height of thecapacity-sensitive feature point and the State of Health is established,as shown in FIG. 10. The coefficient matrix in the linear regressionequation (2) is β=[0.52055, 1.24712E−14]^(T).

In the online estimation stage, the charging voltage, current andtemperature data of a measured battery are firstly recorded when themeasured battery is charged, and then the Incremental Capacity curve isobtained. When the voltage of the temperature-sensitive feature point isreached, the normalized Incremental Capacity curve by standardizedtemperature is obtained according to the Arrhenius fitting functioncalibrated in the offline calibration stage. Then the Box-Coxtransformation is performed on the height of the capacity-sensitivefeature point of the measured battery, and the conversion parameter Ahas been calculated by the offline calibration stage. Finally, thetransformed data is brought into the offline calibrated linearregression equation to estimate the State of Health of the measuredbattery.

The details above are only specific descriptions of the feasibleimplementations of the present invention. They are not used to limit thescope of protection of the present invention. Any equivalent methods ormodifications created by the above technology should be included in theprotection scope of the present invention.

What is claimed is:
 1. An on-line State of Health estimation method fora battery in a wide temperature range comprising the following steps:S1. generating a plurality of incremental capacity curves for a batterybased on a plurality of charging curves for the battery, wherein each ofthe charging curves for the battery is obtained by holding the batteryfor 2 hours at fixed ambient temperatures of −5° C. 0° C. 5° C. 10° C.15° C. 20° C. 25° C. 30° C. 35° C. 40° C. 45° C. 50° C. 55° C. and thencharging each battery with 0.1 C to obtain the charging curve at each ofthe fixed ambient temperatures for the battery; S2. selecting atemperature-sensitive feature point for each of the incremental capacitycurves of the battery at different fixed ambient temperatures, whereinthe temperature-sensitive feature point is a second inflection point ofthe incremental capacity curve, and determining a quantitativerelationship between g voltage shift of the temperature-sensitivefeature point and each fixed ambient temperature other than 25° C. basedon the temperature of the battery at 25° C. by subtracting a voltage atthe temperature-sensitive feature point for the battery at 25° C. from avoltage of the temperature-sensitive feature point of the battery at atemperature other than 25° C. and applying an Arrhenius fitting functionto determine the quantitative relationship from (1): $\begin{matrix}{y = {{a\;{\exp\left( \frac{b}{T} \right)}} + c}} & (1)\end{matrix}$ wherein a, b, c are fitting parameters, T is atemperature, and y represents a voltage shift of temperature-sensitivefeature point; S3. performing a standardized transformation of theincremental capacity curves at different temperatures; and S4.establishing a quantitative relationship between a transformed height ofa capacity-sensitive feature point and a State of Health of the batterybased on a BOX-COX transformation.
 2. The on-line State of Healthestimation method of the battery in the wide temperature range accordingto claim 1, wherein the standardized transformation of the incrementalcapacity curves at different temperatures in S3 comprises obtaining acharging Q-V curve for each battery at the fixed ambient temperaturegreater than 30° C., shifting the charging Q-V curve for each battery atthe fixed ambient temperature greater than 30° C. by a voltage shift toobtain a normalized charging Q-V curve for each battery at the fixedambient temperature greater than 30° C. and obtaining a normalizedincremental capacity curve based on the fixed ambient temperaturegreater than 25° C. through a numerical differentiation method.
 3. Theon-line State of Health estimation method of the battery in the widetemperature range according to claim 2, wherein a corresponding voltageshift of the charging Q-V curve is achieved by bringing the temperatureinto the Arrhenius fitting function.
 4. The on-line State of Healthestimation method of the battery in the wide temperature range accordingto claim 1, wherein the selection of the capacity-sensitive featurepoint in S4 is based on the height of the second inflection point. 5.The on-line State of Health estimation method of the battery in the widetemperature range according to claim 4, wherein based on thecapacity-sensitive feature point, the Box-Cox transformation increaseslinearity between the height of the second inflection point as thecapacity-sensitive feature point and the State of Health using a linearregression equation is expressed as (2): $\begin{matrix}{Y = {{X\;\beta} + ɛ}} & (2)\end{matrix}$ wherein Y is a dependent variable, X is an independentvariable, β is a coefficient matrix, and ε a represents a fitting error;the Box-Cox transformation is represented as: $\begin{matrix}{y_{k}^{(\lambda)} = \left\{ \begin{matrix}\frac{y_{k}^{\lambda} - 1}{\lambda} & {\lambda \neq 0} \\{\ln\; y_{k}} & {\lambda = 0}\end{matrix} \right.} & (3)\end{matrix}$ wherein y in the right side of the equation is an originalvariable, and the subscript k corresponding toy represents a k-thvariable, λ is a conversion parameter, y_(k) ^((λ)) in the left side ofthe equation is a k-th conversion variable.
 6. The on-line State ofHealth estimation method of the battery in the wide temperature rangeaccording to claim 5, wherein the implementation of step 4 alsoincludes; an inverse transformation of is: $\begin{matrix}{y_{k} = \left\{ \begin{matrix}\left( {1 - {\lambda\; y_{k}^{(\lambda)}}} \right)^{1/\lambda} & {\lambda \neq 0} \\{\exp\left( y_{k}^{(\lambda)} \right)} & {\lambda = 0}\end{matrix} \right.} & (4)\end{matrix}$ a maximum likelihood function is used to calculate anoptimal λ, assuming that ε is independent and obeys a normaldistribution, and y conforms to y˜(Xβ,σ²I), X is an independent variablematrix, β is the coefficient matrix, σ² is a variance, I is an identitymatrix, n is number of samples, the maximum likelihood function isexpressed as: $\begin{matrix}{{L\left( {\beta,{\sigma^{2}❘Y},X} \right)} = {{{- \frac{n}{2}}\ln\; 2\pi} - {\frac{n}{2}\ln\;\sigma^{2}} - {\frac{1}{2}{{\sigma^{2}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}} + {\left( {\lambda - 1} \right){\sum\limits_{k = 1}^{n}{\ln\; y_{k}}}}}} & (5)\end{matrix}$ in the formula (5), Y^((λ)) is a dependent variable afterconversion, estimated parameters β and σ² can be expressed as:$\begin{matrix}\left\{ \begin{matrix}{{\beta(\lambda)} = {\left( {X^{T}X} \right)^{- 1}X^{T}Y^{\lambda}}} \\{{\sigma^{2}(\lambda)} = {{\frac{1}{n}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}^{T}\left\lbrack {Y^{(\lambda)} - {X\;\beta}} \right\rbrack}}\end{matrix} \right. & (6)\end{matrix}$ substituting formula (6) into (5) and taking logarithmic:$\begin{matrix}{{L(\lambda)} = {{{- \frac{n}{2}}\left( {{\ln\; 2\pi} + {\ln\;\sigma^{2}} + 1} \right)} + {\left( {\lambda - 1} \right){\overset{n}{\sum\limits_{k = 1}}{\ln\; y_{k}}}}}} & (7)\end{matrix}$ optimal λ is obtained by maximizing formula (7); accordingto the linear regression equation (2), the quantitative relationshipbetween the transformed height of the capacity-sensitive feature pointand the State of Health is then established.
 7. The on-line State ofHealth estimation method of the battery in the wide temperature rangeaccording to claim 1, further comprising generating the state of healthof the battery based on receiving charging voltage, current, andtemperature of a battery to be measured.